Embedded newform 289.2.f.a.103.14

• Label: 289.2.f.a.103.14
• Level: $289$
• Weight: $2$
• Character: 289.103
• Analytic conductor: $2.308$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	289.f (of order \(17\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30767661842\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(24\) over \(\Q(\zeta_{17})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label			103.14
Character	\(\chi\)	\(=\)	289.103
Dual form			289.2.f.a.188.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.149787 - 0.198350i) q^{2} +(-0.219181 - 0.199810i) q^{3} +(0.530419 + 1.86423i) q^{4} +(0.952169 - 0.368872i) q^{5} +(-0.0724630 + 0.0135457i) q^{6} +(-0.380228 - 4.10332i) q^{7} +(0.912759 + 0.353605i) q^{8} +(-0.268689 - 2.89961i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 13 q^{2} - 13 q^{3} - 33 q^{4} - 43 q^{5} - 5 q^{6} - 13 q^{7} - 5 q^{8} - 25 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{4}{17}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.149787	−	0.198350i	0.105915	−	0.140255i	−0.741972	−	0.670431i	\(-0.766109\pi\)
							0.847888	+	0.530176i	\(0.177874\pi\)
\(3\)	−0.219181	−	0.199810i	−0.126544	−	0.115361i	0.607950	−	0.793976i	\(-0.291992\pi\)
							−0.734494	+	0.678615i	\(0.762580\pi\)
\(5\)	0.952169	−	0.368872i	0.425823	−	0.164965i	−0.138787	−	0.990322i	\(-0.544320\pi\)
							0.564611	+	0.825357i	\(0.309026\pi\)
\(7\)	−0.380228	−	4.10332i	−0.143713	−	1.55091i	−0.696482	−	0.717574i	\(-0.745252\pi\)
							0.552769	−	0.833334i	\(-0.313571\pi\)
\(11\)	0.995365	+	3.49835i	0.300114	+	1.05479i	0.954397	+	0.298541i	\(0.0965000\pi\)
							−0.654283	+	0.756250i	\(0.727029\pi\)
\(13\)	6.26375	+	2.42659i	1.73725	+	0.673015i	0.999997	−	0.00260085i	\(-0.000827878\pi\)
							0.737254	+	0.675615i	\(0.236122\pi\)
\(17\)	3.02317	−	2.80365i	0.733225	−	0.679986i
							\(19\)	0.115345	+	0.152742i	0.0264621	+	0.0350414i	0.811056	−	0.584969i	\(-0.198893\pi\)
−0.784594	+	0.620010i	\(0.787128\pi\)
\(23\)	0.606777	+	6.54817i	0.126522	+	1.36539i	0.788117	+	0.615526i	\(0.211056\pi\)
							−0.661595	+	0.749861i	\(0.730120\pi\)
\(29\)	−2.32461	−	8.17014i	−0.431668	−	1.51716i	−0.807161	−	0.590332i	\(-0.798997\pi\)
							0.375492	−	0.926825i	\(-0.377474\pi\)
\(31\)	−3.11408	−	1.20640i	−0.559305	−	0.216676i	0.0649608	−	0.997888i	\(-0.479308\pi\)
							−0.624266	+	0.781212i	\(0.714602\pi\)
\(37\)	−5.05038	−	10.1425i	−0.830277	−	1.66742i	−0.741090	−	0.671406i	\(-0.765691\pi\)
							−0.0891867	−	0.996015i	\(-0.528427\pi\)
\(41\)	1.81439	+	1.65404i	0.283360	+	0.258317i	0.802879	−	0.596142i	\(-0.203301\pi\)
							−0.519519	+	0.854459i	\(0.673889\pi\)
\(43\)	−3.79453	−	2.34947i	−0.578661	−	0.358291i	0.205682	−	0.978619i	\(-0.434059\pi\)
							−0.784343	+	0.620327i	\(0.787000\pi\)
\(47\)	−0.519147	+	5.60249i	−0.0757254	+	0.817207i	0.870174	+	0.492745i	\(0.164007\pi\)
							−0.945899	+	0.324461i	\(0.894817\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.2.f.a.103.14	✓	384
289.188	even	17	inner	289.2.f.a.188.14	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.f.a.103.14	✓	384	1.1	even	1	trivial
289.2.f.a.188.14	yes	384	289.188	even	17	inner